Dear FoMers
Generalization scheme is an FOL axiom scheme that I coined
lately, when added to Extensionality, Impredicative
class comprehension and Pairing it can interpret
set union, power, separation, replacement and
Infinity.
Of course sets are defined as elements of classes.
Generalization scheme: for n=0,1,2,3,....; if phi(x) is
a formula in which z1...zn are the only parameters
in it, then:
[for every z1 is HF ... zn is HF.
(Exist x. x is a class of HF sets and phi(x)) and
for every set x of HF sets. (phi(x) -> x is HF)]
->
[for every set z1...zn. for all x. phi(x) -> x is a set]
is an axiom.
HF stands for the predicate "hereditarily finite" defined below.
a class of HF sets is any class where all its members are HF.
a set of HF sets is any set where all its members are HF.
A weaker version of this scheme is one where "every *class* x
of HF sets" replaces "every *set* x of HF sets" in the above
scheme, it proves all theorems mentioned above except infinity.
Definitions:
The terms subclass and superclass are defined in the standard manner.
A von Neumann ordinal is defined as a transitive class of transitive
sets where every non empty subclass of it has a disjoint element of it.
A natural number is a Von Neumann ordinal that is either empty
or a successor ordinal having every element of it either empty
or a successor ordinal.
A finite class is a class that is have class bijection to some subclass
of a natural number.
The transitive closure of a class is the minimal transitive superclass
of that class.
A hereditarily finite class: is a finite class where every element of
its transitive closure is a finite class.
HF denotes the predicate "a hereditarily finite class".
/
To prove that every HF class is a set: let phi be "x is HF"
To prove Infinity: let phi be "x is a Von Neumann ordinal"
To prove Replacement: let phi be the following formula:
(Exist V. V={y| y is a set} and Exist F:z1-->V and Range(F)=x)
The proofs of set union, powerset and separation are straightforward.
This theory also proves that every natural number defined above is
a finite von Neumann ordinal in the customary sense.
Best Regards
Zuhair